Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory

Muhammad Adnan Samad ^{1, 2, 3}

Yuanqing Xia ^{1, 4}

Saima Siddiqui ^{5, 6}

Muhammad Younus Bhat ⁷

Didar Urynbassarova ⁸

Altyn Urynbassarova ^{9, 10}

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School of automation, Beijing institute of technology, Beijing 100081, China |

Electrical Engineering, Electrical Mechanics and Electrical Technologies Department, Fergana Polytechnic Institute, Fergana 150100, Uzbekistan |

Telecommunication Engineering Department, Tashkent University of Information Technologies, Fergana 150100, Uzbekistan |

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Zhongyuan University of Technology, Zhengzhou 450007, China |

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Department of Mathematics, Fergana Polytechnic Institute, Fergana 150100, Uzbekistan |

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Computer Engineering and Artificial Intelligence Department, Tashkent University of Information Technologies, Fergana 150100, Uzbekistan |

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Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, Awantipora 192122, India

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National Engineering Academy of the Republic of Kazakhstan, Almaty 050010, Kazakhstan |

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Faculty of Information Technology, Department of Information Security, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan |

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Institute of Automation and Information Technologies, Department of Cybersecurity, Information Processing and Storage, Satbayev University, Almaty 050000, Kazakhstan |

Publication type: Journal Article

Publication date: 2025-01-09

MDPI

Journal: Mathematics

scimago Q2

wos Q1

SJR: 0.475

CiteScore: 4.0

Impact factor: 2.3

ISSN: 22277390

DOI: 10.3390/math13020195

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Abstract

The one-dimensional quaternion fractional Fourier transform (1DQFRFT) introduces a fractional-order parameter that extends traditional Fourier transform techniques, providing new insights into the analysis of quaternion-valued signals. This paper presents a rigorous theoretical foundation for the 1DQFRFT, examining essential properties such as linearity, the Plancherel theorem, conjugate symmetry, convolution, and a generalized Parseval’s theorem that collectively demonstrate the transform’s analytical power. We further explore the 1DQFRFT’s unique applications to probabilistic methods, particularly for modeling and analyzing stochastic processes within a quaternionic framework. By bridging quaternionic theory with probability, our study opens avenues for advanced applications in signal processing, communications, and applied mathematics, potentially driving significant advancements in these fields.